Small oscillations of heavy string

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Solution 1

Here is a link that explains how to do it. You need to expand the Lagrangian around the steady solution. That should give you an easier set of differential equations for the small perturbation. Hope this helps.

Solution 2

I was also involved in this problem for the past few weeks.You can write newtons law of motion for small segment of string and obtain a differential equation.From that equation you can find the normal modes of the string and the general motion of the string is given by a superposition of the normal modes.But i ignored the longitudinal oscillations and considered only the transverse motion.The solution was in terms of Bessel function of zeroth order.

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Oiale
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Oiale

Updated on July 09, 2020

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  • Oiale
    Oiale over 3 years

    I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points.

    First of all I wrote down this Lagrangian:

    $$S=\int dt ds \left[\frac{\rho}{2}(\dot{x}^2+\dot{y}^2)-\rho g y(s,t)+\frac{\lambda(s,t)}{2}\left(\left(\frac{\partial x}{\partial s}\right)^2+\left(\frac{\partial y}{\partial s}\right)^2-1\right)\right]$$

    This Lagrangian describes heavy string with fixed ends in gravitational field. Where $\rho$ is density, $g$ is gravitational acceleration, $s$ is natural parameter.

    So I have 3 equations from Euler-Lagrange equations.

    $$\rho\ddot{x}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial x}{\partial s }\right)=0$$ $$\rho\ddot{y}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial y}{\partial s }\right)+\rho g=0$$ $$\left(\frac{\partial x}{\partial s }\right)^2+\left(\frac{\partial y}{\partial s }\right)^2=1$$

    After that I've found stationary solution ($\frac{\partial x}{\partial t}=\frac{\partial y}{\partial t}=\frac{\partial \lambda}{\partial t}=0$). (I just put $\ddot{x}=\ddot{y}=0$)

    $$y_0(x)=-\frac{C_1}{\rho g}\cosh\left(\frac{\rho g x}{C_1}+C_2\right)$$

    Where $C_1,C_2$ is integration constants (depends on positions of ends of string). And $\cosh(x)$ is hyperbolic cosine.

    To study small oscillations I've tried to use pertrubation theory.

    So, I put $$y(s,t)=y_0(s)+\bar{y}(s,t)$$ $$x(s,t)=x_0(s)+\bar{x}(s,t)$$ $$\lambda(s,t)=\lambda_0(s)+\bar{\lambda}(s,t)$$

    But after that I get difficult differential equations, which I can't solve.

    Maybe someone know the more simplier aproach to solve this problem or know how to solve it in this way?